Writing down expressions for traveling waves

1. The problem statement, all variables and given/known data
There is no specific problem - this is more of a broad question...given a wave equation and asked to write down/guess an expression/general solution for a traveling wave, it is sufficient to say the following:

Those may be sufficient guesses if the rest of the context supports them - however, those choices are making assumptions about the boundary conditions and initial values. i.e. what if ##E(0,0)\neq 0## ?

Well that depends if the question is asking for "the" general solutions or if some working guesses would be enough.

Also, the form of the solutions you guessed is fine… but check them against the original equations. There are no ##k## or ##z##, only ##c##. How can you express your "guess" solutions in terms of ##c##?

Just because ##E(0,0)\neq 0##, it does not mean that ##E(x,t)## is a cosine wave.
I'm trying to get you to rethink the assumptions you are making about travelling waves.

Any ##E(x,t)=f(x-ct)##, where ##f## is an arbitrary function, would be your most general possible guess - but it is not all that helpful so you need to use the specifics of the situation to select what sort of ##f## to choose. There is no "best" choice that works for everything.